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Theorem latnlej 14417
Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012.)
Hypotheses
Ref Expression
latlej.b  |-  B  =  ( Base `  K
)
latlej.l  |-  .<_  =  ( le `  K )
latlej.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latnlej  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  -.  X  .<_  ( Y  .\/  Z
) )  ->  ( X  =/=  Y  /\  X  =/=  Z ) )

Proof of Theorem latnlej
StepHypRef Expression
1 latlej.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 latlej.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 latlej.j . . . . . . 7  |-  .\/  =  ( join `  K )
41, 2, 3latlej1 14409 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Y  .<_  ( Y  .\/  Z ) )
543adant3r1 1162 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  .<_  ( Y  .\/  Z
) )
6 breq1 4149 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  ( Y  .\/  Z )  <->  Y  .<_  ( Y 
.\/  Z ) ) )
75, 6syl5ibrcom 214 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  =  Y  ->  X 
.<_  ( Y  .\/  Z
) ) )
87necon3bd 2580 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( -.  X  .<_  ( Y 
.\/  Z )  ->  X  =/=  Y ) )
91, 2, 3latlej2 14410 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Z  .<_  ( Y  .\/  Z ) )
1093adant3r1 1162 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  .<_  ( Y  .\/  Z
) )
11 breq1 4149 . . . . 5  |-  ( X  =  Z  ->  ( X  .<_  ( Y  .\/  Z )  <->  Z  .<_  ( Y 
.\/  Z ) ) )
1210, 11syl5ibrcom 214 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  =  Z  ->  X 
.<_  ( Y  .\/  Z
) ) )
1312necon3bd 2580 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( -.  X  .<_  ( Y 
.\/  Z )  ->  X  =/=  Z ) )
148, 13jcad 520 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( -.  X  .<_  ( Y 
.\/  Z )  -> 
( X  =/=  Y  /\  X  =/=  Z
) ) )
15143impia 1150 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  -.  X  .<_  ( Y  .\/  Z
) )  ->  ( X  =/=  Y  /\  X  =/=  Z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   Latclat 14394
This theorem is referenced by:  latnlej1l  14418  latnlej1r  14419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-lub 14351  df-join 14353  df-lat 14395
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